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diffusion. Compared with the thermodynamic/formal kinetic approaches, the
formalisms of empirical kinetics tend to be more specifically adapted to individual
processes and perhaps less concerned with the proper identification of driving
forces, that is, with the true ''dynamics''. However, we shall attempt to bring out
connections between the two approaches.
3.1.3 Temperature and Pressure Dependence
The rate of change in a system is observed to be not solely dependent on the degree to
which the system departs from equilibrium, as defined in a classical or homogeneous
system by a given variable not being an extremum or in a continuous system by a
variable being nonuniform. The rate may also depend on the values of other variables
even if they are not changing with time themselves. Most notable of these variables is
the temperature. Where the influence of temperature is significant, the process is said
to be thermally activated. Other processes may appear to be athermal, at least to a first
approximation, as in the case of martensitic transformations or crystal plasticity at
relatively low temperatures, but such cases become less common at higher
temperatures. Pressure may also enter as a significant variable influencing the rate of
a process, especially if substantial volume changes are involved.
The dependence of the rate of thermally activated processes on the temperature
can normally be expressed in the exponential form
rate
¼
Ae
RT
ð
3
:
1
Þ
where Q is a parameter with the dimensions of energy per unit amount of sub-
stance, called the activation energy, A is a constant, T is the thermodynamic or
absolute temperature, and R is the gas constant, 8.314 J K
-1
mol
-1
(R
¼
Lk
;
where L is the Avogadro number and k the Boltzmann constant). The symbol E or
E
a
is often used instead of Q for the activation energy and is in many respects
preferable, but Q has been widely used in the materials science literature, espe-
cially for the empirically determined quantity, and is retained here for this reason.
As an empirical expression, the factoring of the multiplier of 1
=
T in (
3.1
) into Q
=
R
is, of course, arbitrary, but it serves to facilitate comparison with theoretically
derived formulae in which kT appears as the fundamental temperature from sta-
tistical thermodynamics.
In a wide range of rate processes, the relation (
3.1
), known as the law of
Arrhenius (
1889
), is found empirically to apply, with constant Q, over a substantial
interval of temperature. This law was originally established empirically, but it can
also be shown to be plausible from a statistical mechanical point of view. How-
ever, the proper theoretical justification of the form (
3.1
) requires a specific
treatment for each process (cf. Flynn
1972
, Chap. 7 for diffusion).
Some properties of the exponential temperature dependence can be seen in the plot
of exp
Q
=
RT
ð
Þ
versus T for different values of Q shown in Fig.
3.1
. Particularly
